Here is another answer which is perhaps more "in the spirit" of the question.
Suppose that $U: \mathcal{A} \to \mathcal{C}$ is a functor of 2-categories. Let $x$ and $y$ be objects of $\mathcal{C}$, and $f: x \to y$ a fixed morphism. Suppose that we have a lift $Y \in \mathcal{A}$ of $y \in \mathcal{C}$, that is an object such that $U(Y) = y$.
We can define the 2-category $Lift(f; Y)$ of lifts of $f$ relative to $Y$. It is the 2-category whose objects are arrows $F: X \to Y$ such that $U(F) = f$, whose morphisms are the obvious triangles (with 2-isomorphisms witnessing the commutativity of the triangle). The top edge of the triangle maps via $U$ to the identity morphism of $x$. The 2-morphisms are maps of the top edges of triangles such that the obvious diagram commutes and such that the 2-morphisms map via $U$ to the identity 2-morphism of the identity 1-morphism of $x$.
I think this is the 2-category of lifts that you meant, or perhaps you meant to restrict just to the invertible part of it. Either way...
Now suppose that:
The functor $U$ is conservative in the following sense: a 1-morphism of $\mathcal{A}$ is an equivalence in $\mathcal{A}$ if and only if its image under $U$ is an equivalence in $\mathcal{C}$; a 2-morphism of $\mathcal{A}$ is an isomorphism if and only if its image under $U$ is an isomorphism; and two 2-morphisms of $\mathcal{A}$ are equal if and only if they are equal after applying $U$. (This is really an analog of being conservative and faithful).
The morphism $f$ is an equivalence.
Then the 2-category $Lift(f; Y)$ is either empty (if no lifts exist) or contractible.
I already gave an example where $\mathcal{A}$ is the category of algebras for a 2-monad where there can fail to be lifts.
However in your situation for coherent 2-groups you know that you can transfer the structure along equivalences. Hence the 2-category $Lift(f; Y)$ is not contractible when $f$ is an equivalence. Moreover we know that the forgetful functor from coherent 2-groups to groupoids is conservative in the sense I describe above. Hence the 2-category $Lift(f; Y)$ of lifts is contractible in this case.